Scale-Up of Powder Mixing

Reference ID: MET-4226 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Pharmaceutical, food and specialty-chemical plants routinely move a new powder blend from a 5–20 kg R&D vessel to a 500–2000 kg production blender. Because the same degree of homogeneity must be reached in the same number of revolutions, reliable scale-up rules are essential. The two most common rules—Froude-number similarity and tip-speed (peripheral-velocity) similarity—trade off mixing time, mechanical stress and power demand. The present sheet implements both options for geometrically similar, free-flowing, non-cohesive powders whose behaviour is dominated by inertial and gravitational forces.

Methodology & Formulas

Constants \[ g = 9.81\ \text{m s}^{-2} \]
Initial Data (lab scale)
mass: \( m_{1} \) density: \( \rho \) impeller diameter: \( D_{1} \) rotational speed: \( N_{1} \)
volume of lab blend: \( V_{b1} = \dfrac{m_{1}}{\rho} \)
Scale-up Geometry
volume ratio: \( \dfrac{m_{2}}{m_{1}} \)
linear scale factor: \( L = \left( \dfrac{m_{2}}{m_{1}} \right)^{1/3} \)
plant-scale diameter: \( D_{2} = L\,D_{1} \)
Lab-Scale Dimensionless Groups
Froude number (sub-critical granular regime): \( Fr_{1} = \dfrac{N_{1}^{2}\,D_{1}}{g} \)
tip speed: \( V_{t1} = \pi\,N_{1}\,D_{1} \)

Similarity Rule (choose one)

Rule	Formula for Plant Speed \( N_2 \)
Fr	\( N_{2} = N_{1}\sqrt{\dfrac{D_{1}}{D_{2}}} \)
Vt	\( N_{2} = \dfrac{V_{t1}}{\pi\,D_{2}} \)

(For any other string the code raises ValueError.)

Plant-Scale Quantities
Froude number: \( Fr_{2} = \dfrac{N_{2}^{2}\,D_{2}}{g} \)
tip speed: \( V_{t2} = \pi\,N_{2}\,D_{2} \)
mixing-time ratio: \( \displaystyle \frac{t_{m2}}{t_{m1}} = \frac{N_{1}}{N_{2}} \)
power-ratio (estimated from fluid-mixing scaling for geometrically similar impellers): \( \displaystyle \frac{P_{2}}{P_{1}} = \left(\frac{N_{2}}{N_{1}}\right)^{3}\left(\frac{D_{2}}{D_{1}}\right)^{5} \) (Note: For powder mixing, this formula is approximate and should be validated empirically.)

Validity Envelope

Check	Threshold
Lab \( Fr_{1} \)	< 2.0
Plant \( Fr_{2} \)	< 2.0
Lab \( V_{t1} \)	< 2.0 m s⁻¹
Plant \( V_{t2} \)	< 2.0 m s⁻¹

Exceeding any limit raises ValueError and indicates that centrifugal segregation (often for Fr > 3) or cohesive/particle-friction effects may become dominant, requiring additional dimensionless groups for accurate scale-up.

Output Set
The sheet returns the rounded values: L, D₂, N₂, Fr₂, V_t2, t_m2/t_m1, and P₂/P₁.

Maintain the same effective volume ratio (bulk powder volume / gross blender volume) used at pilot scale.

Measure the tapped density of the formulation at both scales; recalculate fill level so the headspace percentage above the powder bed is identical.
Verify scale-up with a demonstration batch: collect 10 spatial samples across the blender and target RSD < 3 % for the active or a tracer.
If the industrial blender has a different aspect ratio, consider installing internal baffles or a two-speed sequence (first 50 % speed for loading, then 100 % speed) to recreate the lab-scale shear profile.

Segregation on scale-up usually stems from increased per-particle kinetic energy and longer discharge times.

Match the Froude number (Fr = ω²R/g) between scales to keep similar centrifugal stresses.
Use a mass-flow hopper and eliminate vertical drop heights > 0.5 m during discharge; install socks or valves to control flow.
Add 0.25–0.5 % glidant (e.g., colloidal silica) to reduce inter-particle friction, but confirm final blend uniformity before compression.

Use the constant total revolutions approach rather than constant time.

Scale on revolutions (n) = rotational speed (rpm) × mixing time (min). If 5 min at 20 rpm achieved RSD < 5 % at lab scale (100 revs), target the same 100 revs in production (e.g., 10 min at 10 rpm).
Validate with at least three in-line power draw plateaus; when torque variation < 5 % over 30 s, the endpoint is reached regardless of scale.

Cohesive APIs lose ordered mixing when high shear redistributes fine particles.

Use a pre-blend step: coat 5–10 % of the excipient with 0.5 % magnesium stearate to lower adhesion, then add the API at low shear (< 0.5 kW kg⁻¹) for 2 min.
Scale-up by switching to a split-batch method: prepare several 150 kg sub-batches inside a 600 kg blender using divider plates, then combine and gently mix for 45 s at 20 % of tip speed.
Confirm laser diffraction of the final blend matches the pre-blend PSD within ±3 µm on the D50.

Worked Example: Scale-Up of Powder Mixing

A process engineer must scale up a free-flowing powder mixing process from a laboratory blender to a production-scale blender. The goal is to achieve dynamically similar mixing for a 100-fold increase in batch mass, using Froude number similarity to preserve the granular flow regime.

Knowns (Input Parameters):

Laboratory batch mass, \( m_1 = 10.0 \, \text{kg} \)
Powder bulk density, \( \rho = 500.0 \, \text{kg/m}^3 \)
Laboratory impeller diameter, \( D_1 = 0.250 \, \text{m} \)
Laboratory rotational speed, \( N_1 = 0.800 \, \text{s}^{-1} \)
Plant batch mass, \( m_2 = 1000.0 \, \text{kg} \)
Similarity rule: Constant Froude number (\( Fr \))
Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \)
Mathematical constant \( \pi \approx 3.142 \)

Step-by-Step Calculation:

Calculate the laboratory blender volume: \( V_{b1} = m_1 / \rho \). From the data, \( V_{b1} = 0.020 \, \text{m}^3 \).
Compute the linear geometric scale factor: \( L = (m_2 / m_1)^{1/3} \). Thus, \( L = 4.642 \).
Determine the plant impeller diameter for geometric similarity: \( D_2 = L \times D_1 \). Therefore, \( D_2 = 1.160 \, \text{m} \).
Calculate the laboratory Froude number: \( Fr_1 = N_1^2 D_1 / g \). This gives \( Fr_1 = 0.016 \).
Calculate the laboratory impeller tip speed: \( V_{t1} = \pi N_1 D_1 \). This yields \( V_{t1} = 0.628 \, \text{m/s} \).
Apply Froude similarity to find the plant rotational speed: \( N_2 = N_1 \sqrt{D_1 / D_2} \). Hence, \( N_2 = 0.371 \, \text{s}^{-1} \).
Verify the plant Froude number: \( Fr_2 = N_2^2 D_2 / g \). The result is \( Fr_2 = 0.016 \), confirming similarity with \( Fr_1 \).
Calculate the plant impeller tip speed: \( V_{t2} = \pi N_2 D_2 \). This gives \( V_{t2} = 1.354 \, \text{m/s} \).
Determine the mixing time scale factor: \( t_{m2} / t_{m1} = N_1 / N_2 \). The ratio is \( 2.154 \).
Compute the relative power requirement: \( P_2 / P_1 = (N_2 / N_1)^3 (D_2 / D_1)^5 \). The power ratio is \( 215.443 \).

Final Answer:

For dynamically similar scale-up, the plant-scale blender should have an impeller diameter \( D_2 = 1.160 \, \text{m} \) and operate at a rotational speed \( N_2 = 0.371 \, \text{s}^{-1} \). This maintains a Froude number of \( 0.016 \) and a tip speed of \( 1.354 \, \text{m/s} \). The mixing time will increase by a factor of \( 2.154 \), and the power requirement will be approximately \( 215.4 \) times greater than the laboratory scale.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle